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DISCONTINUOUS HOMOMORPHISMS FROM NON-COMMUTATIVE BANACH ALGEBRAS
Published online by Cambridge University Press: 01 July 1997
Abstract
In the 1970s, a question of Kaplansky about discontinuous homomorphisms from certain commutative Banach algebras was resolved. Let A be the commutative C*-algebra C(Ω), where Ω is an infinite compact space. Then, if the continuum hypothesis (CH) be assumed, there is a discontinuous homomorphism from C(Ω) into a Banach algebra [2, 7]. In fact, let A be a commutative Banach algebra. Then (with (CH)) there is a discontinuous homomorphism from A into a Banach algebra whenever the character space ΦA of A is infinite [3, Theorem 3] and also whenever there is a non-maximal, prime ideal P in A such that [mid ]A/P[mid ]=2ℵ0 [4, 8]. (It is an open question whether or not every infinite-dimensional, commutative Banach algebra A satisfies this latter condition.)
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- © The London Mathematical Society 1997
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